I was thinking the same kind of thing. find an old bike that I can steal the real wheel and derailer from and switch them out. Do I have to switch out the front sprocket, too? I would hold on to the original parts so I can put it back if I ever want to sell it on.
That's exactly what I did, using a 26" Ross 10-speed. If you like to tinker and have done some stuff like this before, it's not too hard. It's a valid point that buying all the Schwinn-correct parts would be expensive, and you could find a 5-speed men's Collegiate and save yourself some work and be riding right away. But I really enjoyed doing this as a summer project, so if taking stuff apart and putting it back together better is your thing, then have at it.
You'll want to use both wheels from the donor bike if it isn't a Schwinn, otherwise the tires won't match since Schwinn used special rims that take tires that won't interchange with other brands. These bikes come with what's called an S-6 rim. Tires for these rims have a 597mm bead diameter, where most others have a 590mm bead diameter, even though they all say 26x1 3/8. It's actually easier to find tires in the 590 size anyway.
The derailleur bolted right on, and I had to spread the frame slightly to get the wider hub with the gear cluster to fit, but I did it pretty easily by hand.
Since the derailleur drivetrain uses a narrower chain, you will need to change the chain and front sprocket also. I just used the whole crankset from the donor bike since the crank arms were the same length. (I actually took off the outer chainring, since the donor was a 10-speed and I prefer the lower gearing anyway.)
On mine, when the bike is in first gear, the chain rubs a little on the chain guard, so I'll have to bend or cut the guard a little. If you look at the guard on your wife's bike, you'll see that it's very similar but has a little spacer at the back so it sits higher.
You can cruise yard sales/flea markets/thrift stores/Craigslist for a 26" 5-speed or 10-speed. Just make sure before taking it home that the wheels are straight and the rear derailleur works. Oh, and you'll need the shifter too.
Keep us posted. I'd love to see someone else doing the same kind of transformation.